Introduction to Matroid Theory
Transcript of Introduction to Matroid Theory
Fundamentals of Matroid
Some Classes of Representable Matroids
Summary
Introduction to Matroid Theory
Congduan Li
Adaptive Signal Processing and Information Theory Research Group
ECE Department, Drexel University
November 21, 2011
Congduan Li Introduction to Matroid Theory
Fundamentals of Matroid
Some Classes of Representable Matroids
Summary
Outline
1 Fundamentals of MatroidGeneral Definition of MatroidsEquivalent DefinitionsOperations
2 Some Classes of Representable MatroidsRepresentable MatroidsExcluded Minors SummaryRelationships between Various Classes of Matroids
3 Summary
Congduan Li Introduction to Matroid Theory
Fundamentals of Matroid
Some Classes of Representable Matroids
Summary
Outline
1 Fundamentals of Matroid
General Definition of Matroids
Equivalent Definitions
Operations
2 Some Classes of Representable Matroids
Representable Matroids
Excluded Minors Summary
Relationships between Various Classes of Matroids
3 Summary
Congduan Li Introduction to Matroid Theory
Fundamentals of Matroid
Some Classes of Representable Matroids
Summary
General Definition of Matroids
Equivalent Definitions
Operations
General Definition of MatroidsVector Independence
Properties of Vector Independence
1 φ is independent to any vector;
2 Hereditary: subset of someindependent vectors are alsoindependent;
3 Augmentation: we can add anew vector to a smallerindependent set to keepindependency.
Example
Let A be the following matrixover field R of real numbers.
A =
1 2 3 4 5 6
1 0 0 1 10 0 1 1 10 1 0 1 0
000
Hereditary: {1, 2, 3} → {1, 2}Augmentation:{1, 2}, {2, 3, 5} → {1, 2, 3}
Congduan Li Introduction to Matroid Theory
Fundamentals of Matroid
Some Classes of Representable Matroids
Summary
General Definition of Matroids
Equivalent Definitions
Operations
General Definition of MatroidsCycles in Graphs
Properties of Graphs
1 φ contains no cycle;
2 Hereditary: subgraph of acycle-free graph is cycle-free;
3 Augmentation: Add a new edgeto a smaller cycle-free subgraph.
Example
Graph associated with A1
54
2
3
6
Example
Let A, B be the followingsubgraphs.
4
13
4
12
6
5
7
Hereditary:{1, 3, 4, 5} → {1, 3, 4}Augmentation:{1, 4}, {1, 3, 4, 5} → {1, 3, 4}
Congduan Li Introduction to Matroid Theory
Fundamentals of Matroid
Some Classes of Representable Matroids
Summary
General Definition of Matroids
Equivalent Definitions
Operations
General Definition of MatroidsCollinear & Coplanar in Geometry
Geometry Diagram
1 Collinear/planar points aredependent (at least 3);
2 φ is not collinear/coplanar toany point;
3 Hereditary: subset of somenon-collinear/coplanar pointsare still non-collinear/coplanar;
4 Augmentation: we can add anew point to a smaller set ofsome non-collinear/coplanarpoints.
Example
See the following geometrydiagram.
1
5
2 43
6
Associated with
A =
1 2 3 4 5 6
1 0 0 1 10 0 1 1 10 1 0 1 0
000
Generalize=⇒Matroid
Congduan Li Introduction to Matroid Theory
Fundamentals of Matroid
Some Classes of Representable Matroids
Summary
General Definition of Matroids
Equivalent Definitions
Operations
General Definition of MatroidsIndependent sets
Definition
A matroid M is an ordered pair (E , I)consisting of a finite set E and acollection I of subsets of E having thefollowing three properties:
1 φ ∈ I;2 Hereditary:
If I ∈ I and I � ⊆ I , then I � ∈ I;3 Augmentation:
If I1 and I2 are in I and |I1| <|I2|, then there is an element e of I2−I1 such that I1 ∪ e ∈ I.
Demo
I is a collection of subset of E
Hereditary
Congduan Li Introduction to Matroid Theory
Fundamentals of Matroid
Some Classes of Representable Matroids
Summary
General Definition of Matroids
Equivalent Definitions
Operations
General Definition of MatroidsIndependent sets
Definition
A matroid M is an ordered pair (E , I)consisting of a finite set E and acollection I of subsets of E having thefollowing three properties:
1 φ ∈ I;2 Hereditary:
If I ∈ I and I � ⊆ I , then I � ∈ I;3 Augmentation:
If I1 and I2 are in I and |I1| <|I2|, then there is an element e of I2−I1 such that I1 ∪ e ∈ I.
Demo
Augmentation
Congduan Li Introduction to Matroid Theory
Fundamentals of Matroid
Some Classes of Representable Matroids
Summary
General Definition of Matroids
Equivalent Definitions
Operations
Outline
1 Fundamentals of Matroid
General Definition of Matroids
Equivalent Definitions
Operations
2 Some Classes of Representable Matroids
Representable Matroids
Excluded Minors Summary
Relationships between Various Classes of Matroids
3 Summary
Congduan Li Introduction to Matroid Theory
Fundamentals of Matroid
Some Classes of Representable Matroids
Summary
General Definition of Matroids
Equivalent Definitions
Operations
Equivalent DefinationsBase/Basis
Definition
Bases B(M) of a matroid are themaximal independent sets.
Properties
Same cardinality for all bases;
The matroid is a span of anybase;
Example
Consider
A =
1 2 3 4 5 6
1 0 0 1 10 0 1 1 10 1 0 1 0
000
Bases: {1,2,3}, {1,2,4},{1,2,5}, {1,3,4}, {1,4,5},{2,3,4}, {2,3,5}, {3,4,5}
Congduan Li Introduction to Matroid Theory
Fundamentals of Matroid
Some Classes of Representable Matroids
Summary
General Definition of Matroids
Equivalent Definitions
Operations
Equivalent DefinationsBase/Basis
Definition
A collection of subsets B ⊆ 2E of aground set E are the bases of amatroid if and only if:
B is non-empty;
base exchange: If B1,B2 ∈ B,for any x ∈ (B1 − B2), there isan element y ∈ (B2 − B1) suchthat (B1 − x) ∪ y ∈ B.
Example
Consider
A =
1 2 3 4 5 6
1 0 0 1 10 0 1 1 10 1 0 1 0
000
Bases: {1,2,3}, {1,2,4},{1,2,5}, {1,3,4}, {1,4,5},{2,3,4}, {2,3,5}, {3,4,5}
Congduan Li Introduction to Matroid Theory
Fundamentals of Matroid
Some Classes of Representable Matroids
Summary
General Definition of Matroids
Equivalent Definitions
Operations
Equivalent DefinitionsCircuit
Definition
The circuits C(M) of a matroid arethe minimal dependent sets.
Property
If any one element is deletedfrom a circuit, it will become anindependent set.
Independent sets I(M) does notcontain any element in C(M).
Consider the graph
1
54
2
3
6
Circuits: {1,2,3,4}, {1,3,5},{2,4,5}, {6}
Congduan Li Introduction to Matroid Theory
Fundamentals of Matroid
Some Classes of Representable Matroids
Summary
General Definition of Matroids
Equivalent Definitions
Operations
Equivalent DefinitionsCircuit
Definition
A collection of sets C is thecollection of circuits of a matroid ifand only if:
φ /∈ Cif C1,C2 ∈ C with C1 ⊆ C2 thenC1 = C2
if C1,C2 ∈ C,C1 �= C2 ande ∈ C1 ∩ C2 then there is someC3 ∈ C with C3 ⊆ (C1∪C2)− e.
Consider the graph
1
54
2
3
6
Circuits: {1,2,3,4}, {1,3,5},{2,4,5}, {6}
Congduan Li Introduction to Matroid Theory
Fundamentals of Matroid
Some Classes of Representable Matroids
Summary
General Definition of Matroids
Equivalent Definitions
Operations
Equivalent DefinitionsRank Function
Definitions
Rank: maximun number of linearlyindependent vectors.For any base of a matroid,r(B) = r(M).Formally, a rank function isr : 2E → N ∪ {0}, for which
For X ⊆ E , 0 ≤ r(X ) ≤ |X |If X ⊆ Y ⊆ E , r(X ) ≤ r(Y )
∀X ,Y ⊆ E , r(X ∩ Y ) + r(X ∪Y ) ≤ r(X ) + r(Y )
Example
Consider
A =
1 2 3 4 5 6
1 0 0 1 10 0 1 1 10 1 0 1 0
000
Bases: {1,2,3}, {1,2,4},{1,2,5}, {1,3,4}, {1,4,5},{2,3,4}, {2,3,5}, {3,4,5};Rank: r(M) = r(B) = 3
Congduan Li Introduction to Matroid Theory
Fundamentals of Matroid
Some Classes of Representable Matroids
Summary
General Definition of Matroids
Equivalent Definitions
Operations
Equivalent DefinitionsClosure
Definitions
Given a rank function of a matroid,the closure operation cl : 2E → 2E isdefined as
cl X = {x ∈ E |r(X ∪ x) = r(X )}.
Closure is a span of a subspace.cl B = E (M);Independent Set:I = {X ⊆ E |x /∈ cl(X − x)∀x ∈ X}
Definition
A subset X of E (M) is said tobe a flat if cl X = X.
Example
Consider
A =
1 2 3 4 5 6
1 0 0 1 10 0 1 1 10 1 0 1 0
000
cl {1,2}={1,2,} → flatcl {1,3}={1,3,5}
Congduan Li Introduction to Matroid Theory
Fundamentals of Matroid
Some Classes of Representable Matroids
Summary
General Definition of Matroids
Equivalent Definitions
Operations
Equivalent DefinitionsClosure
Definitions
A function cl : 2E → 2E is closure fora matroid M = (E , I) iff
If X ⊆ E then X ⊆ clX
If X ⊆ Y ⊆ E , then clX ⊆ clY
If X ⊆ E then cl(cl(X )) = clX
If X ⊆ E and x ∈ E andy ∈ cl(X ∪ x)− cl(X ) thenx ∈ cl(X ∪ y)
Example
Consider
A =
1 2 3 4 5 6
1 0 0 1 10 0 1 1 10 1 0 1 0
000
cl {1,2}={1,2,} → flatcl {1,3}={1,3,5}
Congduan Li Introduction to Matroid Theory
Fundamentals of Matroid
Some Classes of Representable Matroids
Summary
General Definition of Matroids
Equivalent Definitions
Operations
Outline
1 Fundamentals of Matroid
General Definition of Matroids
Equivalent Definitions
Operations
2 Some Classes of Representable Matroids
Representable Matroids
Excluded Minors Summary
Relationships between Various Classes of Matroids
3 Summary
Congduan Li Introduction to Matroid Theory
Fundamentals of Matroid
Some Classes of Representable Matroids
Summary
General Definition of Matroids
Equivalent Definitions
Operations
OperationsDual
Definition
The dual matroid M∗ to a matroidM is the matroid with bases that arecomplements of the bases of M.
B(M∗) = {E − B|B ∈ B(M)}.
(M∗)∗ = M and X is a base ifand only if E − X is a cobase.
Rank function of the dualmatroid M∗:
r∗(X ) = |X |+ r(E − X )− r(M).
Example
1
5
4
632
45
61
23
G G*
Congduan Li Introduction to Matroid Theory
Fundamentals of Matroid
Some Classes of Representable Matroids
Summary
General Definition of Matroids
Equivalent Definitions
Operations
OperationsContraction, Restriction, Minor
Definitions
Restriction of M to X (X ⊂ E ) isthe matroid M � with ground set Xand independent setsI � = {A ∈ I|A ⊂ X}. We denote itas M|X .
Definitions
Deletion: M\T is therestriction of M on E − T ;M\T = M|(E − T )
Example
Congduan Li Introduction to Matroid Theory
Fundamentals of Matroid
Some Classes of Representable Matroids
Summary
General Definition of Matroids
Equivalent Definitions
Operations
OperationsContraction, Restriction, Minor
Definitions
Contraction is the dual operation ofdeletion. M/T = (M∗\T )∗
Definition
Minor: If a matroid M � can beobtained by operatingcombination of restrictions andcontractions on a matroid M,then M � is called a minor of M.
Congduan Li Introduction to Matroid Theory
Fundamentals of Matroid
Some Classes of Representable Matroids
Summary
General Definition of Matroids
Equivalent Definitions
Operations
OperationsExcluded Minor
Definition
Minor-closed matroids are a class ofmatroids of which every minor of amember is also in the class.
Minor-closed classes of matroidshave excluded minorcharacterization, that is,matroids that are not in theclass but have all their properminors in the class.
Example
If a matroid contains U2,4 as aminor, it cannot berepresented by binary vectorspace. U∗
2,4 = U2,4.
Congduan Li Introduction to Matroid Theory
Fundamentals of Matroid
Some Classes of Representable Matroids
Summary
General Definition of Matroids
Equivalent Definitions
Operations
Outline
1 Fundamentals of Matroid
General Definition of Matroids
Equivalent Definitions
Operations
2 Some Classes of Representable Matroids
Representable Matroids
Excluded Minors Summary
Relationships between Various Classes of Matroids
3 Summary
Congduan Li Introduction to Matroid Theory
Fundamentals of Matroid
Some Classes of Representable Matroids
Summary
Representable Matroids
Excluded Minors Summary
Relationships between Various Classes of Matroids
Representable MatroidsBinary
Definition
A matroid that is isomorphic to avector matroid, where all elementstake values from field F, is calledF-representable.
Example
Fano matroid is onlyrepresentable over field ofcharacteristic 2 (Binary).
Binary matroids have excluded minor: U2,4
Congduan Li Introduction to Matroid Theory
Fundamentals of Matroid
Some Classes of Representable Matroids
Summary
Representable Matroids
Excluded Minors Summary
Relationships between Various Classes of Matroids
Representable MatroidsTernary
Definition
Ternary matroids are those who canbe represented over field withcharacteristic 3.
Regular Matroids
Representable over every field
Both Binary and Ternary
Example
U2,k is F-representable if andonly if |F| ≥ k − 1. |U2,4| is3-representable (Ternary).
B =1 2 3 4�1 0 1 −10 1 1 1
�
Ternary matroids have excluded minor: U2,5,U3,5,F7 and F ∗7.
Congduan Li Introduction to Matroid Theory
Fundamentals of Matroid
Some Classes of Representable Matroids
Summary
Representable Matroids
Excluded Minors Summary
Relationships between Various Classes of Matroids
Representable MatroidsGraphic Matroid
Definition
In a graph G = (V ,E ), let C ⊆ 2E
be the sets of the graph cycles. ThenC forms a set of circuits for a matroidwith ground set E . The matroidM(G ) derived in this manner is calleda cycle matroid (Graphic Matroid).
Excluded minors for graphicmatroids:U2,4,F7,F ∗
7,M∗(K5),M∗(K3,3)
Example
K5 matroid and K3,3 matroid:
Figure: K5 matroid
Figure: K3,3 matroid
Congduan Li Introduction to Matroid Theory
Fundamentals of Matroid
Some Classes of Representable Matroids
Summary
Representable Matroids
Excluded Minors Summary
Relationships between Various Classes of Matroids
Representable MatroidsCographic and Planar
Definition
A matroid is called cographic if itsdual is graphic.Ex: M∗(K5) and M∗(K3,3)
Definitions
A both graphic and cographicmatroid is called planar, isomorphicto a cycle matroid derived from aplanar. In a planar, all edges can bedrawn on a plane withoutintersections.
Example
Excluded minors for cographic:U2,4,F7,F ∗
7,M(K5),M(K3,3)
Excluded minors for planar:those excluded minors ofgraphic and cographic
Congduan Li Introduction to Matroid Theory
Fundamentals of Matroid
Some Classes of Representable Matroids
Summary
Representable Matroids
Excluded Minors Summary
Relationships between Various Classes of Matroids
Outline
1 Fundamentals of Matroid
General Definition of Matroids
Equivalent Definitions
Operations
2 Some Classes of Representable Matroids
Representable Matroids
Excluded Minors Summary
Relationships between Various Classes of Matroids
3 Summary
Congduan Li Introduction to Matroid Theory
Fundamentals of Matroid
Some Classes of Representable Matroids
Summary
Representable Matroids
Excluded Minors Summary
Relationships between Various Classes of Matroids
Excluded MinorsVarious Classes of Representable Matroids
Excluded Minors:
Matroid Class Excluded Minor(s)Binary U2,4
Ternary U2,5,U3,5,F7,F ∗7
Regular U2,4,F7,F ∗7
Graphic U2,4,F7,F ∗7,K ∗
5,K ∗
3,3
Cographic U2,4,F7,F ∗7,K5,K3,3
Planar U2,4,F7,F ∗7,K5,K3,3,K ∗
5,K ∗
3,3
Congduan Li Introduction to Matroid Theory
Fundamentals of Matroid
Some Classes of Representable Matroids
Summary
Representable Matroids
Excluded Minors Summary
Relationships between Various Classes of Matroids
Outline
1 Fundamentals of Matroid
General Definition of Matroids
Equivalent Definitions
Operations
2 Some Classes of Representable Matroids
Representable Matroids
Excluded Minors Summary
Relationships between Various Classes of Matroids
3 Summary
Congduan Li Introduction to Matroid Theory
Fundamentals of Matroid
Some Classes of Representable Matroids
Summary
Representable Matroids
Excluded Minors Summary
Relationships between Various Classes of Matroids
Relationships
Relationships between various classes of matroids:
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!!!!!!!!!!!2%#-):+%)#/!
;/3+.%#'7!
<#=&)!>-3/+$&-!?#-&!&%!@&-A?#-&!
>-3/+$&-!?#-&!B!@&-A?#-&!
2%#-):+%)#/!
@&-A9#,,4)!
Figure: Relationships between various classes matroids: U2,4,F7, etc. inthe graph are examples of the matroid class in which they lie. If anexample is not contained in one class, it is called an excluded minor ofthis class.
Congduan Li Introduction to Matroid Theory
Fundamentals of Matroid
Some Classes of Representable Matroids
Summary
Summary
Today’s talk
1 Some Equivalent Definitions of Matroids: Sets, Base, Circuit,Rank, Closure
2 Operations of Matroids: Dual, Restriction, Contraction, Minor
3 Representable Matroids: Binary, Ternary, Regular, Graphic,Cographic, Planar
4 Excluded Minors and relationships between various classes ofmatroids
Congduan Li Introduction to Matroid Theory
Fundamentals of Matroid
Some Classes of Representable Matroids
Summary
Q&A
Thank you!
Next: Matroid, Network and Coding.
James Oxley, Matroid Theory, Oxford University Press, 2011
Congduan Li Introduction to Matroid Theory